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Register a new userDistance Distributions of Cyclic Orbit Codes
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Added by
Hunter Lehmann
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The distance distribution of a code is the vector whose $i^text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $mathbb{F}_{q^n}^*$ on an $mathbb{F}_q$-subspace $U$ of $mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.
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We study orbit codes in the field extension ${mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.
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